Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles

Side rib- is the common side of two adjacent side faces

Prism height- this is a segment perpendicular to the bases of the prism

Prism diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its lateral edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

The bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
Lateral surface - the sum of the areas of all lateral faces of the prism
Total surface - the sum of the areas of all bases and side faces (sum of the area of the side surface and bases)
Side ribs AA 1, BB 1, CC 1 and DD 1.
Diagonal B 1 D
Base diagonal BD
Diagonal section BB 1 D 1 D
Perpendicular section A 2 B 2 C 2 D 2.

Properties of a regular quadrangular prism

The bases are two equal squares
The bases are parallel to each other
The side faces are rectangles
The side edges are equal to each other
The side faces are perpendicular to the bases
The lateral ribs are parallel to each other and equal
Perpendicular section perpendicular to all side ribs and parallel to the bases
Angles of perpendicular section - straight
The diagonal cross section of a regular quadrangular prism is a rectangle
Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " regular quadrangular prism" means that:

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see properties of a regular quadrangular prism above) Note. This is part of a lesson with geometry problems (section stereometry - prism). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To denote the action of extracting the square root in solving problems, the symbol is used√ .

Task.

In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal

√144 = 12 cm.
From where the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm

Answer: 22 cm

Task

Determine the total surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of its side face is 4 cm.

Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 = 4 2
h 2 + 12.5 = 16
h 2 = 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S = 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

Prism. Parallelepiped

Prism is a polyhedron whose two faces are equal n-gons (bases) , lying in parallel planes, and the remaining n faces are parallelograms (side faces) . Lateral rib The side of a prism that does not belong to the base is called the side of the prism.

A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called inclined . Correct A prism is a right prism whose bases are regular polygons.

Height prism is the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. Diagonal section is called a section of a prism by a plane passing through two lateral edges that do not belong to the same face. Perpendicular section is called a section of a prism by a plane perpendicular to the side edge of the prism.

Lateral surface area of a prism is the sum of the areas of all lateral faces. Total surface area is called the sum of the areas of all faces of the prism (i.e. the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism the following formulas are true::

Where l– length of the side rib;

H- height;

S side

S full

S base– area of the bases;

V– volume of the prism.

For a straight prism the following formulas are correct:

Where p– base perimeter;

l– length of the side rib;

H- height.

parallelepiped called a prism whose base is a parallelogram. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called inclined . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped with all edges equal is called cube

The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since a parallelepiped is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of a parallelepiped intersect at one point and bisect it.

2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped the following formulas are valid:

Where l– length of the side rib;

H- height;

P– perpendicular section perimeter;

Q– Perpendicular cross-sectional area;

S side– lateral surface area;

S full– total surface area;

S base– area of the bases;

V– volume of the prism.

For a right parallelepiped the following formulas are correct:

Where p– base perimeter;

l– length of the side rib;

H– height of a right parallelepiped.

For a rectangular parallelepiped the following formulas are correct:

(3)

Where p– base perimeter;

H- height;

d– diagonal;

a,b,c– measurements of a parallelepiped.

The following formulas are correct for a cube:

Where a– rib length;

d- diagonal of the cube.

Example 1. The diagonal of a rectangular parallelepiped is 33 dm, and its dimensions are in the ratio 2: 6: 9. Find the dimensions of the parallelepiped.

Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. by the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Let us denote by k proportionality factor. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. Let us write formula (3) for the problem data:

Solving this equation for k, we get:

This means that the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2. Find the volume of an inclined triangular prism, the base of which is an equilateral triangle with a side of 8 cm, if the side edge is equal to the side of the base and inclined at an angle of 60º to the base.

Solution . Let's make a drawing (Fig. 3).

In order to find the volume of an inclined prism, you need to know the area of its base and height. The area of the base of this prism is the area of an equilateral triangle with a side of 8 cm. Let us calculate it:

The height of a prism is the distance between its bases. From the top A 1 of the upper base, lower the perpendicular to the plane of the lower base A 1 D. Its length will be the height of the prism. Consider D A 1 AD: since this is the angle of inclination of the side edge A 1 A to the base plane, A 1 A= 8 cm. From this triangle we find A 1 D:

Now we calculate the volume using formula (1):

Answer: 192 cm 3.

Example 3. The lateral edge of a regular hexagonal prism is 14 cm. The area of the largest diagonal section is 168 cm 2. Find the total surface area of the prism.

Solution. Let's make a drawing (Fig. 4)

The largest diagonal section is a rectangle A.A. 1 DD 1 since diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of the prism, it is necessary to know the side of the base and the length of the side edge.

Knowing the area of the diagonal section (rectangle), we find the diagonal of the base.

Since then

Since then AB= 6 cm.

Then the perimeter of the base is:

Let us find the area of the lateral surface of the prism:

The area of a regular hexagon with side 6 cm is:

Find the total surface area of the prism:

Answer:

Example 4. The base of a right parallelepiped is a rhombus. The diagonal cross-sectional areas are 300 cm2 and 875 cm2. Find the area of the lateral surface of the parallelepiped.

Solution. Let's make a drawing (Fig. 5).

Let us denote the side of the rhombus by A, diagonals of a rhombus d 1 and d 2, parallelepiped height h. To find the area of the lateral surface of a right parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, because ABCD- rhombus H = AA 1 = h. That. Need to find A And h.

Let's consider diagonal sections. AA 1 SS 1 – a rectangle, one side of which is the diagonal of a rhombus AC = d 1, second – side edge AA 1 = h, Then

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we obtain the equality We obtain the following.

Definition. Prism is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with correspondingly parallel sides, and all edges that do not lie in these planes are parallel.

Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).

All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).

All side faces form lateral surface of the prism .

All lateral faces of the prism are parallelograms .

The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).

Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).

The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .

Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in traversal order, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)

The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)

Among straight prisms, a particular type stands out: regular prisms.

A straight prism is called correct, if its bases are regular polygons.

A regular prism has all lateral faces equal rectangles. A special case of a prism is a parallelepiped.

Parallelepiped

Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.

Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.

Properties and theorems:

Some properties of a parallelepiped are similar to the known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube .All faces of a cube are equal squares. The square of the diagonal is equal to the sum of the squares of its three dimensions

where d is the diagonal of the square;
a is the side of the square.

An idea of a prism is given by:

various architectural structures;
Kids toys;
packaging boxes;
designer items, etc.

The area of the total and lateral surface of the prism

Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's why

S full = S side + 2S main,

Where S full- total surface area, S side-lateral surface area, S base- base area

The lateral surface area of a straight prism is equal to the product of the perimeter of the base and the height of the prism.

S side= P basic * h,

Where S side-area of the lateral surface of a straight prism,

P main - perimeter of the base of a straight prism,

h is the height of the straight prism, equal to the side edge.

Prism volume

The volume of a prism is equal to the product of the area of the base and the height.

Polyhedra

The main object of study of stereometry is spatial bodies. Body represents a part of space limited by a certain surface.

Polyhedron is a body whose surface consists of a finite number of flat polygons. A polyhedron is called convex if it is located on one side of the plane of every plane polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices are vertices of the polyhedron.

For example, a cube consists of six squares, which are its faces. It contains 12 edges (the sides of the squares) and 8 vertices (the tops of the squares).

The simplest polyhedra are prisms and pyramids, which we will study further.

Prism

Definition and properties of a prism

Prism is a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. Polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are lateral edges of the prism.

Prism height is called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-carbon, if its base is an n-gon.

Any prism has the following properties, resulting from the fact that the bases of the prism are combined by parallel translation:

1. The bases of the prism are equal.

2. The lateral edges of the prism are parallel and equal.

The surface of the prism consists of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of the lateral surface of a prism is the sum of the areas of the lateral faces.

Straight prism

The prism is called straight, if its lateral edges are perpendicular to the bases. Otherwise the prism is called inclined.

The faces of a right prism are rectangles. The height of a straight prism is equal to its side faces.

Full prism surface is called the sum of the lateral surface area and the areas of the bases.

With the right prism called a right prism with a regular polygon at its base.

Theorem 13.1. The area of the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, which is the same, by the lateral edge).

Proof. The lateral faces of a right prism are rectangles, the bases of which are the sides of the polygons at the bases of the prism, and the heights are the lateral edges of the prism. Then, by definition, the lateral surface area is:

where is the perimeter of the base of a straight prism.

Parallelepiped

If parallelograms lie at the bases of a prism, then it is called parallelepiped. All faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.

Theorem 13.2. The diagonals of a parallelepiped intersect at one point and are divided in half by the intersection point.

Proof. Consider two arbitrary diagonals, for example, and . Because the faces of a parallelepiped are parallelograms, then and , which means according to To there are two straight lines parallel to the third. In addition, this means that straight lines and lie in the same plane (plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals intersect and are divided in half by the intersection point, which was what needed to be proven.

A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. The lengths of the non-parallel edges of a rectangular parallelepiped are called its linear dimensions (dimensions). There are three such sizes (width, height, length).

Theorem 13.3. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three dimensions (proven by applying Pythagorean T twice).

A rectangular parallelepiped with all edges equal is called cube.

Tasks

13.1 How many diagonals does it have? n-carbon prism

13.2 In an inclined triangular prism, the distances between the side edges are 37, 13 and 40. Find the distance between the larger side edge and the opposite side edge.

13.3 A plane is drawn through the side of the lower base of a regular triangular prism, intersecting the side faces along segments with an angle between them. Find the angle of inclination of this plane to the base of the prism.

Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Right prism
Theorem 2. The area of the lateral surface of the prism

Parallelepiped:
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped

Prism is a polyhedron whose two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than the bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Lateral ribs edges that do not belong to the bases are called. The union of lateral faces is called lateral surface of the prism, and the union of all faces is called the full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. Straight prism called a prism whose side ribs are perpendicular to the planes of the bases. Correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.

Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P^ - perpendicular section perimeter;
S b - lateral surface area;
V - volume;
S p is the area of the total surface of the prism.

V=SH
S p = S b + 2S o
S b = P ^ l

Definition 1 . A prismatic surface is a figure formed by parts of several planes parallel to one straight line, limited by those straight lines along which these planes successively intersect one another*; these lines are parallel to each other and are called edges of the prismatic surface.
*It is assumed that every two successive planes intersect and that the last plane intersects the first

Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same is true for triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to AC) like the line of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC is equal to A"C"), like opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.

Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.

Definition 3 . A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to the prismatic surface - side faces; edges of the prismatic surface - side ribs of the prism. By virtue of the previous theorem, the base of the prism is equal polygons. All lateral faces of the prism - parallelograms; all side ribs are equal to each other.
Obviously, if the base of the prism ABCDE and one of the edges AA" in size and direction are given, then it is possible to construct a prism by drawing edges BB", CC", ... equal and parallel to edge AA".

Definition 4 . The height of a prism is the distance between the planes of its bases (HH").

Definition 5 . A prism is called straight if its bases are perpendicular sections of the prismatic surface. In this case, the height of the prism is, of course, its side rib; the side edges will be rectangles.
Prisms can be classified according to the number of lateral faces equal to the number of sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.

Theorem 2 . The area of the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be a given prism and abcde its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. The face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that coincides with ab; the area of the face ВСВ "С" is equal to the product of the base ВВ" by the height bc, etc. Consequently, the side surface (i.e. the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", ВВ", .., for the amount ab+bc+cd+de+ea.